Robustness of Structurally Equivalent Concurrent Parity Games

TL;DR

This paper investigates the robustness and continuity of the value function in concurrent and turn-based stochastic parity games, providing tight bounds and demonstrating the necessity of structural equivalence for value stability.

Contribution

It introduces quantitative bounds on value differences due to transition imprecision and proves value continuity and strategy robustness under structural equivalence assumptions.

Findings

Value difference bounds are tight and asymptotically optimal.

Value continuity holds for structurally equivalent games.

Robustness of strategies is shown for turn-based games under structural equivalence.

Abstract

We consider two-player stochastic games played on a finite state space for an infinite number of rounds. The games are concurrent: in each round, the two players (player 1 and player 2) choose their moves independently and simultaneously; the current state and the two moves determine a probability distribution over the successor states. We also consider the important special case of turn-based stochastic games where players make moves in turns, rather than concurrently. We study concurrent games with \omega-regular winning conditions specified as parity objectives. The value for player 1 for a parity objective is the maximal probability with which the player can guarantee the satisfaction of the objective against all strategies of the opponent. We study the problem of continuity and robustness of the value function in concurrent and turn-based stochastic parity gameswith respect to imprecision in the transition probabilities. We present quantitative bounds on the difference of the value function (in terms of the imprecision of the transition probabilities) and show the value continuity for structurally equivalent concurrent games (two games are structurally equivalent if the support of the transition function is same and the probabilities differ). We also show robustness of optimal strategies for structurally equivalent turn-based stochastic parity games. Finally we show that the value continuity property breaks without the structurally equivalent assumption (even for Markov chains) and show that our quantitative bound is asymptotically optimal. Hence our results are tight (the assumption is both necessary and sufficient) and optimal (our quantitative bound is asymptotically optimal).